Total Variation Depth for Functional Data

TL;DR

This paper introduces the total variation depth, a new measure for functional data that effectively detects outliers by decomposing into shape and magnitude components, with theoretical analysis and practical demonstrations.

Contribution

It proposes the total variation depth for functional data, including a decomposition into shape and magnitude, and demonstrates its effectiveness for outlier detection and visualization.

Findings

Effective outlier detection for functional data.

Decomposition into shape and magnitude components.

Successful application to real datasets.

Abstract

There has been extensive work on data depth-based methods for robust multivariate data analysis. Recent developments have moved to infinite-dimensional objects such as functional data. In this work, we propose a new notion of depth, the total variation depth, for functional data. As a measure of depth, its properties are studied theoretically, and the associated outlier detection performance is investigated through simulations. Compared to magnitude outliers, shape outliers are often masked among the rest of samples and harder to identify. We show that the proposed total variation depth has many desirable features and is well suited for outlier detection. In particular, we propose to decompose the total variation depth into two components that are associated with shape and magnitude outlyingness, respectively. This decomposition allows us to develop an effective procedure for outlier detection and useful visualization tools, while naturally accounting for the correlation in functional data. Finally, the proposed methodology is demonstrated using real datasets of curves, images, and video frames.